Four velocity, time component

I am having a little trouble to understand how do we find the time component of the four velocity of a particle in a Schwarzschild metric space, in its rest frame:
I mean, the time component is not defined as $dx^{0}/dtau$?

In the Schwarzschild metric we have: $$ds^2 = -f dt^2 + f^{-1} dr^2 + r^2 d Omega ^2.$$

Shouldn’t $x^{0} = dt/dtau = f^{-1/2}$ ? It was give to me that it is equal to $f^{-1}$, why?